Question

To obtain the area of a sector, what fraction is multiplied by the area of a circle (A = πr2)?

Answer 1

Let r be a radius of a given circle and α be an angle, that corresponds to a sector.

The circle area is $A=\pi r^2$ and denote the sector area as $A_1$. 
Then  $\dfrac{A_1}{A}= \dfrac{\alpha}{2\pi}$  (the ratio between area is the same as the ratio between coresponding angles).

$A_1=\dfrac{\alpha}{2\pi} \cdot A=\dfrac{\alpha}{2\pi} \cdot \pi r^2= \dfrac{r^2\alpha}{2}$.